Page 1 Math 251 Fall 2015 copyright Joe Kahlig 1 False Lines could be skew False only unit if the angle between the vectors is 90 degrees True True 2 Hyperboloid of one sheet x2 4y 2 2z 2 10 ellipsoid 2x2 2y 2 z 2 50 circular Paraboloid 5y 30x2 30z 2 hyperbolic cylinder 4y 2 6z 2 24 cone 2x2 4y 2 z 2 0 3 complete the square center 3 5 0 and radius 48 4 Let the second vector be h0 0 1i now use the dot product formula to get 3 cos 14 a c c 5 projc a c 2 1 2 1 6 6 6 6 compute a scalar triple product of these vectors a b c to get that the volume is 226 7 The direction vector for the line is v1 h2 1 7i A vector from the point 1 3 4 to the point 3 3 5 is v2 h2 0 1i The normal vector for the plane is v1 v2 h1 12 2i Answer x 12y z 29 8 r0 t h3 cos t 4 3 sin ti and r0 t a L b s R5 1 Rt 25 5 r0 t dt 20 r0 u du 0 Rt 5du 5t 0 s 4s s r s 3 sin 3 cos 5 5 5 3 4 3 r0 t cos t sin t c T t 0 r t 5 5 5 0 T t 3 d 0 r t 25 9 Since the line is perpendicular to the plane this means the direction vector of the line is parallel to the normal vector of the plane so let the direction vector be some scalar multiple of the normal vector v n h3 2 4i Answer x 1 3t y 1 2t z 1 4t 10 an equation of the line is x 1 4t y t and z 2 plug this into the formula for the surface and solve for t you get t 1 and t 3 points 5 1 2 and 13 3 2 Check the back of the page for more problems Math 251 Fall 2015 copyright Joe Kahlig Page 2 11 t 2 gives the point so the tangent vector direction vector is v f 0 2 h2 16 12i x 5 2t y 16 16t z 8 12t 8 r0 t r00 t 12 r0 t 3 1 8t2 3 13 Method 1 If there is a plane P2 containing the line L where P2 is parallel to the given plane P then the normal vector of P and the normal vector of P2 are parallel Since the line lies in P2 then the dot product of the direction vector of the line and the normal vector of P2 will be zero In similar fashion the dot product of the direction vector of L and plane P will also be zero The direction vector of the line is v h4 2 1i and normal vector of plane P is n h1 3 1i v n 9 6 0 so the answer is no there is not a plane containing line L that will be parallel to plane P Method 2 If there is a plane P2 containing the line L where P2 is parallel to the given plane P then the line will not intersect the plane So check to see if there is a value of t will the line will intersect the plane x 3y z 23 1 4t 3 1 2t 1 t 23 lots of work t 2 so the line intersects the plane so it is not parallel to the plane Check the back of the page for more problems
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